Definition 1.1 
A function \(q \colon {+n} \rightarrow  is called a \emph {Herglotz-Nevan\-linna function} if it is holomorphic and has non-negative imaginary part.\) 
This is a well-studied class of functions, appearing e.g. in [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. Particularly when considering such functions of one variable, they appear in many areas of complex and functional analysis and numerous applications. Some examples of these include the moment problem [10]}, [11]}, [12]}, the theory of Sturm-Liouville problems and perturbations [13]}, [14]}, [15]}, [16]}, when deriving physical bounds for passive systems [17]} or as approximating functions in certain convex optimization problems [18]}, [19]}. Applications concerning functions of several variables appear e.g. when considering operator monotone functions [1]} or with representations of multidimensional passive systems [9]}.
